The space Mor_d^N of degree d morphisms f from P^N to itself is parameterized by the space of coefficients of the polynomials that define f. This space has a natural structure as an affine variety. Elements R of Aut(P^N), or equivalently of PGL_{N+1}, act on Mor_d^N via conjugation, f --> R^{-1} o f o R. Conjugation commutes with iteration, so it is the natural action to consider when studying the dynamics of f, since conjugate maps determine the same dynamical system up to simultaneous change of variables of the domain and range. In this talk I will discuss how geometric invariant theory yields a nice quotient Mor_d^N//Aut(P^N), how this quotient can be embedded in various larger spaces with even nicer properties, what happens when one adds dynamical level structure, some structure theorems and open problems, and, as time permits, various related topics such as field of moduli versus field of definition problems for dynamical systems.